Introduction

Predator-prey relationships can be important for conservation, including the interactions between tigers and their ungulate prey. There is a good deal of ecological theory about these relationships but not much hard evidence from the field. We don’t (yet) have much data on tigers.

However, at Isle Royale NP, Michigan, which is an isolated island in Lake Superior, the moose and wolf populations and kill rates have been monitored continuously since 1971. The project has a wonderful web site with wolf photos and a research blog here.

Vucetich et al (2002) used these data to test a range of theoretical models in which kill rates are related to prey density, predator density, or the ratio between them. We will look at just three of the best models.

We will do this in Excel, and we will see each stage of the process in calculating model parameters, likelihoods, and AICs (Akaike’s Information Criterion) for each model. Comparing AICs showed which of the theoretical models best fits the situation on Isle Royale.

MS Excel® and the "Solver" add-in

In Excel go to the 'Tools' menu and look for 'Solver...' . If it is not there, click on 'Add-ins...' and make sure that "Solver Add-in" is checked. You may need to install the add-ins, for which you will require the original MS Office installation discs.

Other spreadsheet programs such as StarOffice and OpenOffice, may eventually have equivalent functionality, but not at the time of writing. Note that 'Goal Seek' in Excel, StarOffice and OpenOffice is not equivalent to 'Solver'.

Excel is useful for this kind of exercise and the results have been cross-checked with those from statistical software. However, Excel is not a statistical package and I don't advise you to use it as the sole means of analyzing your own data. See here for some of the problems with statistical analysis in spreadsheet packages.

Working through the analysis

Download the file "wolf_kill_rates.xls".

Open the file and check the ‘Prey dependent’ worksheet. Estimates of moose (N) and wolf (P) abundances are given for each year. In most years, there were several packs of wolves, and separate kill rates (K) are given for each pack. The kill rate is expressed as number of kills per wolf per month.

Download the lab guide "wolf_kill_rates.pdf". You will probably want to print out the lab guide and have it next to your computer while you work through the instructions.

Work through the lab guide before checking the results below. Enjoy!

Results

Vucetich et al (2002) found that the best model out of the 14 they tested was the ratio-dependent model that you have fitted to the data. The predator-dependent model ranked third with deltaAICc = 1.3, and the prey-dependent model was ninth, deltaAICc = 20.9.

If those results don't tally with those you got, there is a model answer here.

Main points

• A model is a precise mathematical expression of a theoretical idea which gives an ‘expected’ or ‘predicted’ value for comparison with the observed value. Models usually involve one or more parameters.
   
• The difference between observed and predicted values is used to calculate the likelihood of parameter values, so Maximum Likelihood Estimates can be found.
   
• When maximized, the likelihood is used to calculate AIC (or AICc or similar statistic). AIC can be used to compare different models, provided the same data are used.
   
• The best model has the lowest AIC. Models within 2 units of the best are also good candidates, ie. there is uncertainty about which is the best.

What next?

• Check out the other models tried by Vucetich et al (2002) - the equations are given in Table 1 of their paper.
   
• Try running the models in R with the 'optim' function: data files and an R script are here.
   
• The book on ecological modeling is Hilborn & Mangel (1997)